Geometry and Measurement of Plane Figures Activity Set 4 · GEOMETRY AND MEASUREMENT OF PLANE...

Geometry and Measurement of Plane Figures

Activity Set 4

Trainer Guide

GEOMETRY AND MEASUREMENT OF PLANE FIGURES—AcTIvITY SET 4 Int_PGe_04_TGCopyright© by the McGraw-Hill Companies—McGraw-Hill Professional Development

GEOMETRY AND MEASUREMENT OF PLANE FIGURES—AcTIvITY SET 4 Int_PGe_04_TG

Copyright© by the McGraw-Hill Companies—McGraw-Hill Professional Development 1

GEOMETRY AND MEASUREMENT OF PLANE FIGURESACTIvITY SET #4

NGSSS 3.G.3.1 NGSSS 3.G.3.3 NGSSS 4.G.5.1 NGSSS 5.G.3.1

Amazing Angles

In this activity, participants explore angle concepts in polygon shapes.

Materials

• Transparency/Page:AngleTypes• Transparency/Page:MeasuringAngles• Transparency/Page:ACircleofMeasure• Transparency/Page:PolygonAnglesChart• Transparency/Page:PolygonAnglesChart

AnswerKey• plain 3 5 cards (4 per participant)• rulerforeachparticipant• protractorforeachparticipant• scissorsforeachparticipant• pens/pencils(multicolorpens)• blanktransparency

Vocabulary

• degree• angle• vertex• rightangle• straightangle

tiMe:30minutes




INTRODUCE

•Remindparticipantsthatoneaspectofgeometryistheapplicationofanglestovariousshapesandfigures.

•DisplayTransparency:AngleTypes.

•Goovertheangledescriptionsandnames.

teaching tip: It may help to clarify the definitions if you explain the meaning of adjacent—having a common side or border and, in mathematics, a common endpoint.

•DisplayTransparency:MeasuringAngles and have participantstakeouttheirmatchingpages.

•Takeoutaprotractor.

•Demonstrateonangle1howtomeasureanangle.

◆ Alignthe0˚markandlinewiththeright-handsideoftheangle,makingsurethatthevertexoftheangleisalignedwiththecentermarkofthe0˚line.(Thereisusuallyasmallholeatthislocationtoenableyoutoplacethevertexappropriately.)

◆ Locatetheleft-handsideoftheangleandtracethelinetothedegreemarkontheprotractor.

•Haveparticipantsmeasuretheremaininganglesandwritethedegreesthattheyfindintheappropriateblanksontheirpages.

•Goovertheanswerswiththeparticipantsanddemonstratethemeasurementprocess,ifnecessary,toaddressanyquestions.

•Reviewthedefinitionsatthebottomofthepage.

interior angle an angle formed by two sidesof a polygon

adjacent angles angles that share a commonside and a common vertexbetween them, but that donot share any interior points

exterior angle an angle adjacent to, butoutside of, a polygon–formedby extending one side of thepolygon

angle types

McGraw-Hill Professional Development GEOMETRY AND MEASUREMENT OF PLANE FIGURES/31

A

B

C A

B

C

A

B

D

C

interior exterior adjacent

Transparency: Angle Types

acute angle—an angle less than 90º

obtuse angle—an angle more than 90º

right angle—an angle equal to 90º

straight angle—an angle of 180º

Measuring angles


1

45

23

angle 2 angle 3angle 1

angle 5angle 4

Transparency: Measuring Angles




teaching tip: If time permits, have participant volunteers come to the front to measure the angles and record the results on the transparency.

teaching tip: If the group is advanced, have them also identify the angle type after they measure.

• 1—acuteangle• 2—acuteangleandadjacentangle(adjacentto

angle 3)• 3—obtuseangleandadjacentangle(adjacentto

angle2)• 4—straightangle(straightline)• 5—rightangle(formedbyperpendicularlines)

Ask why none of the angles are interior or exterior. (Theyarenotpartof,oradjacentto,polygons.)

•Askparticipantshowmanydegreesarearoundthecenterofacircle.

•DisplayTransparency:ACircleofMeasure.

•Pointouttoparticipantsthatthedistancearound thecenterofthecircle(360˚)isthebasisforallanglemeasure.

•Pointoutonthetransparencythatthediameterofthecircle(astraightline)dividestherevolutioninhalf,creatingastraightangle,or180˚.

•Explaintoparticipantsthattheywillusethisinformationtohelpthemfindthenumberofdegreesintheinterioranglesofatriangle.

It is a mathematics convention that the unit ofangle measure (degree) is 1

360 of a completerevolution around the center of a circle.

a circle of Measure


Transparency: A Circle of Measure




DISCUSS AND DO

•Distributetoeachparticipantfour3 5 cards and a pairofscissors.

•Haveeachparticipantdrawononecardalarge triangle.

teaching tip: Have participants use a straightedge, ruler, or card side to draw all figures. Straightedges are required to achieve accuracy for the activity. Also, no shape that they create can have overlapping edges.

•Haveparticipantscutouttheirtriangles.

•Haveparticipantsuseapenorpenciltocolorintheanglesabout1

2outfromeachvertex.

•Havethemcutthetriangleinto3pieces,witheachpiececontaining1angle.

•Tellthemtolaythe3anglestogetherwiththeverticesjoiningandtheirsidestouching.

•Pointoutthattheynowhaveastraightlineorastraightangle,whichisdefinedas180˚.

•Pointoutthattheyallmadedifferentkindsoftriangles.

•Explainthattheanglesofalltrianglessumto180˚.

•Haveparticipantsusetheirrulerstodrawanothertriangleononeoftheir3 5 cards.

•Havethemmakethetrianglesaslargeaspossibleforeaseofmeasurement.

•Tellthemtomeasureeachangleintheirtriangles,usingtheirprotractors,andaddthethreeanglestogether.




•Displayablanktransparency.

•Havevariousvolunteerparticipantssharetheanglemeasureswithintheirtriangles.

•Recordontheblanktransparencytheanglemeasuresas they are shared.

•Pointoutthattheanglesdifferedindividually, butthatthesumoftheanglesforanytriangle wasalways180˚.

•Haveparticipantstakeouttheirthirdcards.

•Pointoutthatthecardisarectangle.

•Askthemtocolorthe4cornersandcutthecardinto4pieces(1cornertoeachpiece).

•Askthemtoarrangethecornerstogetherandtell youhowmanydegreesthereareintheanglesof arectangle.(360˚)

•Explainthatanyquadrilateralhasanglesthatsum to360˚.

•Drawarectangleonablanktransparency.

•Drawadiagonalfromonecorneroftherectangletothe corner opposite.

•Pointoutthattheanglesofthe2trianglesthusformedalsosumto360˚.

•Haveparticipantsdrawa5-sidedfigureontheir last cards.

•Havethemcolorthecornerangles,cutouttheshape,andthencutitinto5pieces—1angleperpiece.

•Askparticipantstolaytheanglestogetherinsuchawaythattheycantellyouthetotalnumberofdegrees.

•Suggestthat,whenparticipantscomplainthattheycannotmatchalltheangles,theycreatemorethanonefigure.




•Askparticipantshowmanydegreesthereareintheanglesofapentagon.(540˚)

•Haveoneparticipantcomeupwithhisorhershapesandillustrateontheoverheadprojectorhisorhersolution.

•Drawa5-sidedpolygononablanktransparency.

•Draw,from1vertex,linestoallopposingverticesforwhichyoucanmaketriangles.

•Pointoutthattheanglesofthe3trianglesthusformedalsosumto540˚.

CONCLUDE

•DisplayTransparency:PolygonAnglesChartand have participantstakeouttheirmatchingpages.

•Fillin,alongwiththeparticipants,thefirstthreerowsofthePolygonAnglesChartusinginformationthattheyhavecollectedduringthisactivity.

•Encourageparticipantstocreatetrianglesofeachshapetohelpthem.

•Askparticipantsthenumberofdegreesthattheythinktheanglesofahexagonwouldtotal.(720˚)

•Completethehexagonrowonthechart.

•Askparticipantsiftheyrecognizeapattern. (Theruleis(n–2)•180˚.)

•Askparticipantshowthisruleisderived. (n,thenumberofsides,less2isthenumber ofnon-overlappingtrianglesineachshape.)

GEOMETRY AND MEASUREMENT OF PLANE FIGURES—AcTIvITY SET 4 TRANS_K6_PG_04Copyright© 2002 by the McGraw-Hill Companies—McGraw-Hill Professional Development

triangle

quadrilateral

pentagon

hexagon

heptagon

octagon

nonagon

decagon

Polygon Number of Number of Sides Number of Degrees: Name Sides Minus 2

polygon angles chart

Transparency: Polygon Angles Chart




•Writetheruleonthetransparencyinthefourthcolumnheading.

•Godowntothelastfiguresonthechart.

•Askparticipantsforthenumberofdegreesat eachrow.

•Fillinthetransparencyateachstep.

•RefertoTransparency:PolygonAnglesChartAnswerKey, as necessary.

End of Amazing AnglesGEOMETRY AND MEASUREMENT OF PLANE FIGURES—AcTIvITY SET 4 TRANS_K6_PG_04Copyright© 2002 by the McGraw-Hill Companies—McGraw-Hill Professional Development

n n – 2 (n – 2) • 180°

3

4

5

6

7

8

9

10

1 180°

2 360°

3 540°

4 720°

5 900°

6 1,080°

7 1,260°

8 1,440°

triangle

quadrilateral

pentagon

hexagon

heptagon

octagon

nonagon

decagon


polygon angles chart Answer Key

Transparency: Polygon Angles Chart Answer Key




Race to Place

Inthisactivity,participantsusegeometricknowledgethattheyremembertomatchpicturesofanglesandshapeswiththeirdefinitions.

Materials

• Transparency/Page:RacetoPlaceDirections• Transparency/Page:TriangleFactsAnswerKey• Transparency/Page:AngleFactsAnswerKey• Transparency/Page:AnglesinShapesAnswerKey• Transparency/Page:LineFactsAnswerKey• Transparency/Page:CircleFactsAnswerKey• RacetoPlaceCards• 5pocketcharts• bell

tiMe:15minutes

teaching tip: Post the pocket charts with their definitions before the beginning of the activity. Use the Facts transparencies as a guide for the definitions that go with each title. Space the charts around the room with a lot of room between them.




INTRODUCE

•Suggesttoparticipantsthatovertimetheyhaveaccumulatedalotofknowledgeaboutthewaylines,shapes,andangleswork.

•Pointoutthefivechartsandtheirdefinitions.

•Explaintotheparticipantsthattheywillcompeteasteamstomatchgeometricdefinitionswithpicturesthatillustratetheconceptsdefined.

teaching tip: If you have a large group, assign pairs instead of single people to each card.

DISCUSS AND DO

•DisplayTransparency:RacetoPlaceDirections.

•Gooverthestepsofthegame.

•Haveparticipantsmoveinto4or5equal-sizedgroups.

•Distributetheshapecards—allofonecoloredshape toeachgroup,onecardperperson.

•Call,“Go.”

•Havethefirstgrouptofinishsendonemembertothefrontoftheroomtoringthebell.

teaching tip: If a team member cannot place his or her shape card, he or she should go to the end of the line and wait to place the card after other team members have placed their cards.

teaching tip: If the group is inexperienced, permit them a few moments to look at the definition sheets(AnswerKeys)beforethegame.

GEOMETRY AND MEASUREMENT OF PLANE FIGURES—AcTIvITY SET 4 TRANS_K6_PG_04Copyright© 2002 by the McGraw-Hill Companies—McGraw-Hill Professional Development

Directions• Distribute your team cards evenly among the members

of your team.

• Have team members play their cards in relay fashion.

• Have a player:

• race to the chart that holds the definition of the picture on his or her card

• place the card next to the definition

• race back to the team and sit down

• Have the next person race to the chart and place his orher card.

• Have one team member race to the front and ring thebell when all the team’s cards are correctly placed.

race to place

Transparency: Race to Place Directions




CONCLUDE

•Congratulatetheparticipantsforbeingabletoremembersomanygeometryconceptsanddefinitions.

•DisplaytheAnswerKeytransparenciesinturn,quicklyreviewingthedefinitions.

•Emphasizethefollowingdefinitionsforeachkey:

◆ Transparency:TriangleFactsAnswerKey –equilateraltriangle –righttriangle(especiallyhypotenuse)

◆ Transparency:AngleFactsAnswerKey –straight angle –vertical angles

◆ Transparency:AnglesinShapesAnswerKey –triangle –equilateraltriangle

◆ Transparency:LineFactsAnswerKey –alternate interior angles

◆ Transparency:CircleFactsAnswerKey –circumference

End of Race to Place


• A scalene triangle has no congruent sides and no congruent angles.

• An isosceles triangle has 2 congruent sides and 2 congruent angles.

• An equilateral triangle has 3 congruent sides and 3 congruent angles.

• The angles of an acute triangle areall less than 90˚.

• One angle in an obtuse triangle is greater than 90˚.

• A right triangle has one angle equal to 90˚. The side opposite the 90˚ angle is called the hypotenuse.

triangle Facts Answer Key

Transparencies: Triangle Facts Answer Key, Angle Facts Answer Key, Angles in Shapes Answer Key, Line Facts Answer Key, Circle Facts Answer Key

GEOMETRY AND MEASUREMENT OF PLANE FIGURES—AcTIvITY SET 4 Int_PGe_04_PMCopyright© by the McGraw-Hill Companies—McGraw-Hill Professional Development

interior angle an angle formed by two sides of a polygon

adjacent angles angles that share a common side and a common vertex between them, but that do not share any interior points

exterior angle an angle adjacent to, but outside of, a polygon—formed by extending one side of the polygon

Angle Types

A

B

C A

B

C

A

B

D

C

interior exterior adjacent


acute angle—an angle less than 90º

obtuse angle—an angle more than 90º

right angle—an angle equal to 90º

straight angle—an angle of 180º

Measuring Angles

1

45

23

angle 2 angle 3angle 1

angle 5angle 4


It is a mathematics convention that the unit of angle measure (degree) is 1

360 of a complete revolution around the center of a circle.

A Circle of Measure


triangle

quadrilateral

pentagon

hexagon

heptagon

octagon

nonagon

decagon


Polygon Angles Chart


n n – 2 (n – 2) • 180°

3

4

5

6

7

8

9

10

1 180°

2 360°

3 540°

4 720°

5 900°

6 1,080°

7 1,260°

8 1,440°

triangle

quadrilateral

pentagon

hexagon

heptagon

octagon

nonagon

decagon


Polygon Angles Chart Answer Key


Directions •Distributeyourteamcardsevenlyamongthemembersofyourteam.

•Haveteammembersplaytheircardsinrelayfashion.

•Haveaplayer:

• racetothechartthatholdsthedefinitionofthe pictureonhisorhercard

• placethecardnexttothedefinition

• racebacktotheteamandsitdown

•Havethenextpersonracetothechartandplacehisorhercard.

•Haveoneteammemberracetothefrontandringthebellwhenalltheteam’scardsarecorrectlyplaced.

Race to Place


• Ascalenetrianglehasnocongruentsidesandnocongruentangles.

• Anisoscelestrianglehas2congruentsidesand2congruentangles.

• Anequilateral trianglehas3congruentsidesand3congruentangles.

• Theanglesofanacute trianglearealllessthan90˚.

• Oneangleinanobtuse triangleisgreaterthan90˚.

• Aright trianglehasoneangleequalto90˚.Thesideoppositethe90˚angleiscalledthehypotenuse.

Triangle Facts Answer Key


• Anacute angleislessthan90˚.

• Anobtuse angleisgreaterthan90˚andlessthan180˚.

• Astraight angleisequalto180˚.

• Aright angleisequalto90˚.

• Anglesthatshareacommonsidebetweenthemareadjacent.

• Twoanglesthatsumto180˚arecalledsupplementary.

• Nonadjacentanglesformedbytwointersectinglinesarecalledvertical angles.Theyhavethesamemeasure.

Angle FactsAnswer Key


• Atrianglehasanglesthatsumto180˚.

• Arectanglehasanglesthatsumto360˚.

• Anglesinsideashapeareinterior angles.

• Anglesoutsideashapeareexterior angles.

• Thebaseanglesandoppositesidesofanisoscelestrianglearecongruent.

• Thesidesandanglesofanequilateraltrianglearecongruent.

Angles in Shapes Answer Key


• Asetofpoints,astraightpath,thatextendsindefinitelyin2oppositedirectionsisaline.

• Aline segmentis2endpointsandthestraightpathbetweenthem.

• Perpendicularlinesformrightangles.

• Ifalineintersectstwoparallellines,thealternate interior anglesareequal.

• Parallellinesareequidistantfromeachother.

Line FactsAnswer Key

6 cm6 cm


• Acompleterevolutionaroundthecenterofacirclehas360º.

• Achordisalinesegmentthatconnectstwopointsonthecircumferenceofacircle.

• Thelinesegmentjoiningthecenterofthecircleandapointonitscircumferenceiscalledaradius.

• Adiameter isachordthatpassesthroughthecenterofacircle.Itslengthistwicethatoftheradiusofthecircle.

• Acircleisthesetofallpointsinaplanethatareequidistantfromaspecifiedpoint.

• Thedistancearoundacircleiscalleditscircumference.

Circle Facts Answer Key


GlossaryGeometry and Measurement of Plane Figures

angle Geometricfiguremadeof2raysor2linesegmentsthatsharethesameendpoint,calledavertex.

area Thenumberofsquareunitsinaregion.

congruent Havingthesameshape,size,and/ormeasure.

degree Aunitformeasuringangles.

irregular polygon Apolygoninwhichnotallthesidesarecongruentand/ornotalltheangleshavethesamemeasure.

line Asetofpointsformingastraightpathin2directionsthatareoppositeeachother.

perimeter Thedistancearoundtheoutsideofashapeorfigure.

plane Aflatsurfacethatextendsforeverinalldirections.

point Alocationinspace.

polygon Aclosedshapemadeupofaminimumof3linesegments.

quadrilateral Apolygonwith4sides.

rectangle Aquadrilateralwith4rightangles.


regular polygon Apolygoninwhichallthesidesarecongruentandalltheangleshavethesamemeasure.

triangle Apolygonwith3sides.

Glossary (continued)


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A complete revolution around the center of a

circle has 360º.

A chord is a line segment that connects two points

on the circumference of a circle.

The line segment joining the center of the circle and

a point on its circumference is called a radius.


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Ifalineintersectstwoparallellines,the

alternate interior angles areequal.

Parallel linesareequidistantfromeachother.

Aright angle isequalto90°.


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Asetofpointsthatextendindefinitelyin2oppositedirectionsisaline.

Aline segment hastwoendpoints.

Perpendicular linesformrightangles.


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Anglesoutsideashapeareexterior angles.

Thebaseanglesandoppositesidesofan

isosceles trianglearecongruent.

Thesidesandanglesofanequilateral triangle

arecongruent.


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Atriangle hasanglesthatsumto180˚.

Arectangle hasanglesthatsumto360˚.

Anglesinsideashapeareinterior angles.


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Non adjacent angles formed by two intersecting

lines are called vertical angles. They have the

same measure.

Angles that share a common side between

them are adjacent.

Two angles that sum to 180˚ are called

supplementary.


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Anacute angle islessthan90˚.

Anobtuse angle isgreaterthan90˚andlessthan180˚.

Astraight angle isequalto180˚.


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Theanglesofanacute triangle areall

lessthan90˚.

Oneangleinanobtuse triangle isgreaterthan90˚.

Aright triangle hasoneangleequalto90˚.


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Thediameterisachordthatpassesthroughthecenterofacircle.

Acircle isthesetofallpointsinaplanethatareequidistantfromaspecifiedpoint.

Thedistancearoundacircleiscalleditscircumference.


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Ascalene triangle hasnocongruentsidesandnocongruentangles.

Anisosceles triangle has2congruentsidesand2congruentangles.

Anequilateral triangle has3congruentsidesand3congruentangles.


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Geometry and Measurement of Plane Figures Activity Set 4 · GEOMETRY AND MEASUREMENT OF PLANE...

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